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In a previous paper, we proposed a heterotic dark energy model, called$\Lambda$HDE, in which dark energy is composed of two components: cosmologicalconstant (CC) and holographic dark energy (HDE). The aim of this work is togive a more comprehensive and systematic investigation on the cosmologicalimplications of the $\Lambda$HDE model. Firstly, we make use of the currentobservations to constrain the $\Lambda$HDE model, and compare its cosmology-fitresults with the results of the $\Lambda$CDM and the HDE model. Then, bycombining a qualitative theoretical analysis with a quantitative numericalstudy, we discuss the impact of considering curvature on the cosmic evolutionsof fractional HDE density $\Omega_{hde}$ and fractional CC density$\Omega_{\Lambda}$, as well as on the ultimate cosmic fate. Finally, we explorethe effects of adopting different types of observational data. We find that:(1) the current observational data cannot distinguish the $\Lambda$HDE modelfrom the $\Lambda$CDM and the HDE model; this indicates that DE may containmultiple components. (2) the asymptotic solution of $\Omega_{hde}$ and thecorresponding cosmic fate in a flat universe can be extended to the case of anon-flat universe; moreover, compared with the case of a flat universe,considering curvature will make HDE closer to a phantom dark energy. (3)compared with JLA dataset, SNLS3 data more favor a phantom type HDE; incontrast, using other types of observational data have no significant impact onthe cosmic evolutions of the $\Lambda$HDE model.

A closer look at the cosmological implications of the $\Lambda$HDE model